Solving a Problem with Non-Constant g: Expanding a Maclaurin Series

[didinyc]

Homework Statement

Suppose an object is dropped from height h above Earth where h<<R, but is large enough so g, the acceleration due to gravity, is NOT constant! Show that speed with which it hits the ground, neglecting friction, is approximately, v= sqrt2gh *(1-(h/2R))

Hint: you will need to expand an expression in Maclaurin series.

The Attempt at a Solution

I tried to use conservation of energy and find g by using the gauss' law i got a close result but i didnt get the 2 in the last solution! Then I realize I cannot use conservation of energy because the g is not constant so i cannot use mgh! any ideas how could i solve this proble

 

[tiny-tim]

welcome to pf!

hi didinyc! welcome to pf!

… because the g is not constant so i cannot use mgh!

so use mMG/(r+h)

 

[didinyc]

hi didinyc! welcome to pf! so use mMG/(r+h)

thank you! I tried what you said too but the answer that i got is wrong.It is really close answer except the 2 in front of the R in the v equation! so this is what i got as my result: v= sqrt2gh *(1-(h/R))!

 

[tiny-tim]

hi didinyc!

(just got up :zzz: …)

(have a square-root: √ )

show us your full calculations, and then we'll see what went wrong!

 

[rwisz]

m?

There are a few different approaches you could take to solve this problem. One option is to use the Maclaurin series expansion of the gravitational potential energy, which can account for non-constant g. This would involve finding the first few terms of the series and using them to calculate the potential energy at different heights. From there, you could use the conservation of energy principle to find the speed at which the object hits the ground.

Another approach could be to use the Taylor series expansion of the acceleration due to gravity, which can also account for non-constant g. This would involve finding the first few terms of the series and using them to calculate the acceleration at different heights. From there, you could use the equations of motion to find the speed at which the object hits the ground.

Regardless of the approach you choose, it would be important to carefully consider the assumptions and limitations of your calculation, such as neglecting friction and assuming a small h in relation to the Earth's radius. You may also want to compare your result to the solution for a constant g to see how accurate your approximation is.